Spurs are listening to offers for the Wales international, who is ready to leave the Lilywhites in January.
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The Daily Mail says Palace are determined to keep the former Nottingham Forest winger in London, knowing Bournemouth, Aston Villa and Everton are also keen.
Palace have now opened talks with Spurs for Johnson, with a price of around £35m being negotiated.
For their part, Spurs will seek to use the funds raised by Johnson's sale to strengthen manager Thomas Frank's attacking options over the coming weeks.